Sen. Jon Kyl, R-Ariz., and Sen. Kay Bailey Hutchison, R-Texas, unveil their bill dubbed the "Achieve Act." It allows illegal immigrants brought to the USA as children to remain and apply for citizenship over time. / Alex Wong, Getty Images

by Alan Gomez, USA TODAY

by Alan Gomez, USA TODAY

WASHINGTON -- Republicans and Democrats emerged from the fall elections agreeing on at least one thing: that it was finally time for both sides to come together to fix the nation's broken immigration system.

But judging from the first week of legislative action on immigration, figuring out how to do that will be a contentious battle between, and within, the two parties.

On the Republican side, Sens. Jon Kyl of Arizona and Kay Bailey Hutchison of Texas introduced a bill that grants illegal immigrants brought to the country as children a series of visas to remain in the USA for years. That bill, known as the "Achieve Act," requires immigrants to go to the back of the line to apply for green cards and, eventually, U.S. citizenship, a requirement not supported by Democrats and immigration advocates.

The senators' pitch was followed Thursday by debate on the House floor on a proposal from GOP Rep. Lamar Smith of Texas that increases the number of visas for highly skilled immigrants. Smith's bill would cut a diversity visa program that brings in 55,000 people a year from mostly African countries, another requirement opposed by Democrats and immigration advocates.

"Incredibly tone-deaf," said Frank Sharry, executive director of America's Voice, a group that supports the legalization of the country's illegal immigrants. "What the election has shown is there's real juice, there's real power behind the movement to win citizenship for immigrants who are Americans in all but paperwork."

Sharry was referring to the success that President Obama had with Hispanics, winning that vote 71% to 27% over Republican challenger Mitt Romney, Pew Hispanic Center statistics show.

Mark Krikorian, executive director of the Center for Immigration Studies, which opposes widespread legalization, said Republicans need to reach a compromise on the issue of young illegal immigrants to show they're "serious and engaged in the debate." But, he said, interpreting the election results as a mandate to legalize all illegal immigrants is "laughable."

"I understand the tone-deaf story line, but it's baloney," he said. "Republicans just aren't doing what the Democrats want them to do."

The Congressional Hispanic Caucus - a group of 21 Democratic members of Congress - on Wednesday outlined their priorities, which included a pathway to citizenship for the nation's illegal immigrants and more visas for families, highly skilled immigrants and "guest workers." All of that, they believe, should be done in one sweeping bill.

There is disagreement even within Republican circles on how to approach immigration bills during the lame-duck session and in the new Congress that convenes in January.

Some conservatives, like Sen. Marco Rubio, R-Fla., say it's best to tackle the problem piecemeal. Rubio has been drafting legislation to address the young illegal immigrant population. From there, he would tackle border security, ways to screen job applicants to ensure they're in the country legally and the refinement of the nation's guest-worker program, which allows non-citizens to work in the U.S. for a while before returning home. Only when that is finished would he address the fate of the nation's illegal immigrants, Rubio spokesman Alex Conant said.

Some in his party disagree with that strategy.

"We have a window of opportunity to fix this problem, and I think it has to be fixed in its entirety," said Rep. Mario Diaz-Balart, R-Miami, who has been meeting with members of both parties to craft a solution.

Many Democrats, including President Obama, agree on that point. The White House issued a statement Wednesday saying it supports some aspects of Smith's bill for highly skilled immigrants. But the administration made clear it opposed the bill partly because it did not address the immigration system as a whole.

"The administration does not support narrowly tailored proposals that do not meet the president's long-term objectives with respect to comprehensive immigration reform," the statement read.

The road to an immigration solution likely will be rocky, but many are encouraged that the debate is taking place. Rep. Luis Gutierrez, D-Ill., has been working on immigration legislation for years and remembers a time when his Democratic colleagues were uncomfortable tackling the subject. He recalled caucus meetings where he and a few others were the lone voices calling for action.

"We stood alone sometimes in the corner," he said. "All of the sudden, we're the belle of the ball. We're here to say it's time to start the dance."